Weyl law for 1-cycles

TL;DR

This paper proves Gromov's conjectured Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds, extending previous methods with new parametric inequalities and approximation techniques.

Contribution

It establishes the Weyl law for 1-cycle volume spectrum using parametric coarea and isoperimetric inequalities, including a novel version for families of 0-cycles.

Findings

Weyl law confirmed for 1-cycles in n-manifolds

Development of parametric inequalities for families of cycles

Introduction of approximation methods with δ-localized families

Abstract

We prove the Weyl law for the volume spectrum for $1$-cycles in $n$-dimensional manifolds which was conjectured by Gromov. We follow the strategy of Guth and Liokumovich of obtaining the Weyl law from parametric versions of the coarea inequality and the isoperimetric inequality. A version of the later for families of $0$-cycles is shown in this article. We also obtain approximation results by $\delta$-localized families, which are used to prove the parametric inequalities.